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Notions of computability at higher types II
 In preparation
, 2001
"... ntroduce some simple general theory to allow us to talk about notions of highertype computable functional. The following definitions (with minor variations) appear frequently in the literature. Definition 1.1 (Weak partial type structures) A weak partial type structure, or weak PTS A [over a set X ..."
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ntroduce some simple general theory to allow us to talk about notions of highertype computable functional. The following definitions (with minor variations) appear frequently in the literature. Definition 1.1 (Weak partial type structures) A weak partial type structure, or weak PTS A [over a set X], consists of the following data: . for each type #, a set A # of elements of type # [equipped with a canonical bijection A 0 # = X], . for each #, # , a partial application function ## : A ### A # # A # . We usually omit type subscripts from application operations, and often write x y simply as xy. By convention, w
Linear Programming Languages
"... Abstract. We formalize SℓPCF, namely a programming language which is able to represent linear function between coherence spaces. We give an interpretation of SℓPCF into the model of linear coherence spaces and we show that such semantics is fully abstract with respect to it. SℓPCF is not syntactical ..."
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Abstract. We formalize SℓPCF, namely a programming language which is able to represent linear function between coherence spaces. We give an interpretation of SℓPCF into the model of linear coherence spaces and we show that such semantics is fully abstract with respect to it. SℓPCF is not syntactically linear, namely its programs can contain the same variable more than once. Last, we address the universality problem. 1
Universality results for models in locally Boolean domains
 IN COMPUTER SCIENCE LOGIC
, 2006
"... In [6] J. Laird has shown that an infinitary sequential extension of PCF has a fully abstract model in his category of locally boolean domains (introduced in [8]). In this paper we introduce an extension SPCF ∞ of his language by recursive types and show that it is universal for its model in locall ..."
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In [6] J. Laird has shown that an infinitary sequential extension of PCF has a fully abstract model in his category of locally boolean domains (introduced in [8]). In this paper we introduce an extension SPCF ∞ of his language by recursive types and show that it is universal for its model in locally boolean domains. Finally we consider an infinitary target language CPS ∞ for (the) CPS translation (of [16]) and show that it is universal for a model in locally boolean domains which is constructed like Dana Scott’s D ∞ where D = 1
A Proof System for Correct Program Development
, 2000
"... realworld applications (e.g. [EHM + 99, Buh95]). Moreover, aspects of ML such as strong typing and the exceptions system have significantly influenced the design of languages such as Java [GJS96], and it seems likely that future systems languages will incorporate many of these features [Mac00]. ..."
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realworld applications (e.g. [EHM + 99, Buh95]). Moreover, aspects of ML such as strong typing and the exceptions system have significantly influenced the design of languages such as Java [GJS96], and it seems likely that future systems languages will incorporate many of these features [Mac00]. Regarding the second requirement, even before the definition of ML had fully taken shape, the LCF system [GMW78] provided a program logic for a rather restricted fragment of the language. Subsequent research has sought to build on the definition in order to support formal reasoning about programs. Most notably, the Extended ML project [KST97] resulted in a formal language for specifying program properties, but the complexity of this language prohibited the development of useful proof rules. A di#erent approach has been pursued by Elsa Gunter et al [GV94], who have formalized the definition of ML within the HOL theorem prover; this has proved useful for metatheo
The elimination of nesting in SPCF
 In Proceedings of TLCA ’05, number 3461 in LNCS
, 2005
"... Abstract. We use a fully abstract denotational model to show that nested function calls and recursive definitions can be eliminated from SPCF (a typed functional language with simple nonlocal control operators) without losing expressiveness. We describe — via simple typing rules — an affine fragmen ..."
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Abstract. We use a fully abstract denotational model to show that nested function calls and recursive definitions can be eliminated from SPCF (a typed functional language with simple nonlocal control operators) without losing expressiveness. We describe — via simple typing rules — an affine fragment of SPCF in which function nesting and recursion (other than iteration) are not permitted. We prove that this affine fragment is fully expressive in the sense that every term of SPCF is observationally equivalent to an affine term. Our proof is based on the observation of Longley — already used to prove universality and full abstraction results for models of SPCF — that every type of SPCF is a retract of a firstorder type. We describe retractions of this kind which are definable in the affine fragment. This allows us to transform an arbitrary SPCF term into an affine one by mapping it to a firstorder term, obtaining an (affine) normal form, and then projecting back to the original type. In the case of finitary SPCF, the retraction is based on a simple induction, which yields bounds for the size of the resulting term. In the infinitary case, it is based on an analysis of the relationship between SPCF definable functions and strategies for computing them sequentially. 1
Matching typed and untyped realizability (Extended abstract)
"... Realizability interpretations of logics are given by saying what it means for computational objects of some kind to realize logical formulae. The computational objects in question might be drawn from an untyped universe of computation, such as a partial combinatory algebra, or they might be typed ob ..."
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Realizability interpretations of logics are given by saying what it means for computational objects of some kind to realize logical formulae. The computational objects in question might be drawn from an untyped universe of computation, such as a partial combinatory algebra, or they might be typed objects such as terms of a PCFstyle programming language. In some instances, one can show that a particular untyped realizability interpretation matches a particular typed one, in the sense that they give the same set of realizable formulae. In this case, we have a very good fit indeed between the typed language and the untyped realizability model—we refer to this condition as (constructive) logical full abstraction. We give some examples of this situation for a variety of extensions of PCF. Of particular interest are some models that are logically fully abstract for typed languages including nonfunctional features. Our results establish connections between what is computable in various programming languages, and what is true inside various realizability toposes. We consider some examples of logical formulae to illustrate these ideas, in particular their application to exact realnumber computability. The present article summarizes the material I presented at the Domains IV workshop, plus a few subsequent developments; it is really an extended abstract for a projected journal paper. No proofs are included in the present version. 0
Realizability Models for Sequential Computation
, 1998
"... We give an overview of some recently discovered realizability models that embody notions of sequential computation, due mainly to Abramsky, Nickau, Ong, Streicher, van Oosten and the author. Some of these models give rise to fully abstract models of PCF; others give rise to the type structure of seq ..."
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We give an overview of some recently discovered realizability models that embody notions of sequential computation, due mainly to Abramsky, Nickau, Ong, Streicher, van Oosten and the author. Some of these models give rise to fully abstract models of PCF; others give rise to the type structure of sequentially realizable functionals, also known as the strongly stable functionals of Bucciarelli and Ehrhard. Our purpose is to give an accessible introduction to this area of research, and to collect together in one place the definitions of these new models. We give some precise definitions, examples and statements of results, but no full proofs. Preface Over the last two years, researchers in various places (principally Abramsky, Nickau, Ong, Streicher, van Oosten and the present author) have come up with a number of new realizability models that embody some notion of "sequential" computation. Many of these give rise to fully abstract and universal models for PCF and related languages. Alth...
Topic F of APPSEM
"... ped strategy (in the sense of the AJM model of Abramsky et. al.) for d. Accordingly, the realisability model over A eff wb , the effective wellbracketed strategies, is even universal in the sense that all elements of the model appear as denotations of PCF terms. In a future version of [9] there ..."
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ped strategy (in the sense of the AJM model of Abramsky et. al.) for d. Accordingly, the realisability model over A eff wb , the effective wellbracketed strategies, is even universal in the sense that all elements of the model appear as denotations of PCF terms. In a future version of [9] there will also be included the discussion of other pca's of gametheoretic nature giving rise to fully abstract and universal models which originate from S. Abramsky's work on game semantics for classical linear logic. Independently, fully abstract realisability models for PCF have been constructed by Marz, Rohr and Streicher in [14] using as underlying pca's term models for untyped calculus with arithmetic. In this case the proof is not via the AJM game model but instead makes use of the category SD of sequential domains as described in [13] originating from a reformulation and generalisation of [18]. As describ
Part II Local Realizability Toposes and a Modal Logic for
"... 5.1 Definition and Examples 5.1.1 Definition and Definability Results A tripos is a weak tripos with disjunction which has a (weak) generic object. Explicitly we define: ..."
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5.1 Definition and Examples 5.1.1 Definition and Definability Results A tripos is a weak tripos with disjunction which has a (weak) generic object. Explicitly we define: